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\newcommand{\eZ}{\euf{Z}} \newcommand{\U}{\mbf{1}} \newcommand{\F}{{\mrm{F}}} \newcommand{\G}{\Gamma} \newcommand{\dG}{\digamma} \newcommand{\DG}{{\mrm{I}\hspace{-0.3ex}\G}} \newcommand{\ucF}{{\ul{\cF\!}\,}} \newcommand{\uG}{{\ul{\G\!}\,}} \newcommand{\uDG}{{\ul{\DG\!}\,}} \newcommand{\I}{{I\!\!\!\;I}} \newcommand{\f}{\text{\scriptsize$\sF$}} \newcommand{\s}{\text{\scriptsize$\sS$}} \newcommand{\st}{\text{\scriptsize$\sT$}} \newcommand{\sst}{\text{\tiny$\sT$}} \newcommand{\bplus}{\text{\,\footnotesize$\boxplus$\,}} %%Code for 'theorem', definitions and abbreviations \newcommand{\bex}{\begin{example}} \newcommand{\eex}{\end{example}} \newcommand{\bexer}{\begin{xca}} \newcommand{\eexer}{\end{xca}} \newcommand{\bdefi}{\begin{definition}} \newcommand{\edefi}{\end{definition}} \newcommand{\brem}{\begin{remark}} \newcommand{\erem}{\end{remark}} \newcommand{\bthm}{\begin{theorem}} \newcommand{\ethm}{\end{theorem}} \newcommand{\bprop}{\begin{proposition}} \newcommand{\eprop}{\end{proposition}} \newcommand{\bcor}{\begin{corollary}} \newcommand{\ecor}{\end{corollary}} \newcommand{\blem}{\begin{lemma}} \newcommand{\elem}{\end{lemma}} \newcommand{\PROOF}{\begin{proof}} \newcommand{\QED}{\end{proof}} \begin{document} \setcounter{page}{16} \noindent\parbox{2.85cm}{\includegraphics*[keepaspectratio=true,scale=1.75]{BJMA.jpg}} \noindent\parbox{4.85in}{\hspace{0.1mm}\\[1.5cm]\noindent Banach J. Math. Anal. 3 (2009), no. 2, 16--27\\ $\frac{\rule{4.55in}{0.05in}}{{}}$\\ {\footnotesize \textcolor[rgb]{0.65,0.00,0.95}{\textsc{\textbf{\large{B}}anach \textbf{\large{J}}ournal of \textbf{\large{M}}athematical \textbf{\large{A}}nalysis}}\\ ISSN: 1735-8787 (electronic)\\ \textcolor[rgb]{0.00,0.00,0.84}{\textbf{http://www.math-analysis.org }}\\ $\frac{{}}{\rule{4.55in}{0.05in}}$}\\[.5in]} \title[$E_0$--Semigroups for Product Systems: The Nonunital Case]{$E_0$--Semigroups for Continuous Product Systems:\\The Nonunital Case} \author[M.\ Skeide]{Michael Skeide} \address{Universit\`a\ degli Studi del Molise, Dipartimento S.E.G.e S., Via de Sanctis, 86100 Campobasso, Italy.} \email{\textcolor[rgb]{0.00,0.00,0.84}{skeide@unimol.it}} \dedicatory{{\rm Communicated by M. Frank}} \subjclass[2000]{Primary 46L55; Secondary 46L53, 46L08.} \keywords{Quantum probability, quantum dynamic, product system, Hilbert module, classification.} \date{Received: 13 January 2009; Accepted: 7 April 2009.} \begin{abstract} Let $\cB$ be a \nbd{\sigma}unital \nbd{C^*}algebra. We show that every strongly continuous \nbd{E_0}semigroup on the algebra of adjointable operators on a full Hilbert \nbd{\cB}module $E$ gives rise to a full continuous product system of correspondences over $\cB$. We show that every full continuous product system of correspondences over $\cB$ arises in that way. If the product system is countably generated, then $E$ can be chosen countably generated, and if $E$ is countably generated, then so is the product system. We show that under these countability hypotheses there is a one-to-one correspondence between \nbd{E_0}semigroups up to stable cocycle conjugacy and continuous product systems up to isomorphism. This generalizes the results for unital $\cB$ to the \nbd{\sigma}unital case. \end{abstract} \maketitle \vspace{1in} \end{document} .